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A327831 Numbers m such that sigma(m)*tau(m) is a square but sigma(m)/tau(m) is not an integer. 1
232, 2152, 3240, 3560, 3944, 6516, 17908, 22504, 23716, 26172, 32360, 34344, 36584, 37736, 43300, 45612, 48204, 55080, 55912, 60520, 61480, 69352, 73084, 78184, 79056, 79300, 96552, 104168, 105832, 106088, 125356, 130432, 133864, 140040, 149992, 163764, 168424, 172840, 176360, 183204 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

If sigma(m)/tau(m) is a square (m is in A144695) then sigma(m)*tau(m) is also a square (m is in A327830), but the converse is false (see 232 in the Example section). This sequence consists of these counterexamples.

It seems that all terms are even. - Marius A. Burtea, Oct 15 2019

LINKS

Table of n, a(n) for n=1..40.

EXAMPLE

sigma(232) = 450 and tau(232) = 8, so sigma(232)*tau(232) = 450*8 = 3600 = 60^2 and sigma(232)/tau(232) = 450/8 = 225/4 is not an integer, hence 232 is a term.

MAPLE

filter:= u -> sigma(u)/tau(u) <> floor(sigma(u)/tau(u)) and issqr(sigma(u)*tau(u)) : select(filter, [$1..100000]);

MATHEMATICA

sQ[n_] := IntegerQ@Sqrt[n]; aQ[n_] := sQ[(d = DivisorSigma[0, n]) * (s = DivisorSigma[1, n])] && !sQ[s/d]; Select[Range[2*10^5], aQ] (* Amiram Eldar, Oct 15 2019 *)

PROG

(PARI) isok(m) = my(s=sigma(m), t=numdiv(m)); issquare(s*t) && (s % t); \\ Michel Marcus, Oct 15 2019

(MAGMA) [k:k in [1..200000] | not IsIntegral(a/b) and IsSquare(a*b) where a is DivisorSigma(1, k) where b is #Divisors(k)]; // Marius A. Burtea, Oct 15 2019

CROSSREFS

Equals A144695 \ A327830.

Similar to A327624 with sigma(m) and phi(m).

Cf. A003601 (sigma(m)/tau(m) is an integer), A023883 (sigma(m)/tau(m) is an integer and m is nonprime).

Cf. A000005 (tau), A000203 (sigma).

Sequence in context: A234690 A234683 A238919 * A234682 A156391 A279660

Adjacent sequences:  A327828 A327829 A327830 * A327832 A327833 A327834

KEYWORD

nonn

AUTHOR

Bernard Schott, Oct 14 2019

STATUS

approved

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Last modified January 18 21:06 EST 2021. Contains 340262 sequences. (Running on oeis4.)